The magnetic spherical pendulum is a mechanical system consisting of a pendulum whereof the bob is electrically charged, moving under the influence of gravitation and the magnetic field induced by a magnetic monopole deposited at the origin. Physically not directly realizable, it turns out to be equivalent to a reduction of the Lagrange top. This work is essentially the logbook of our attempts at understanding the simplest contemporary approaches to the magnetic spherical pendulum.

Thesis (Ph. D.)--University of Texas at Austin, 2000. / Vita. Includes bibliographical references (leaves 165-167). Available also in a digital version from Dissertation Abstracts.

Thesis (doctoral)--University of Stockholm, 1984. / Abstract (1 leaf) inserted. Bibliography: p. 79-80.

Thesis (Ph.D.)--University of California, Berkeley, 1961. / "UC-34 Physics" -t.p. "TID-4500 (16th Ed.)" -t.p. Includes bibliographical references (p. 102).

Thesis (Ph.D.) - University of Queensland, 2005. / Includes bibliography.

Thesis (Ph. D.)--University of Washington, 2000. / Vita. Includes bibliographical references (p. 120-124).

No description available.

Thesis (MSc)--University of Stellenbosch, 2006. / ENGLISH ABSTRACT: We are studying a possible implementation of an appropriate framework for a proper non- Hermitian quantum theory. We present the case where for a non-Hermitian Hamiltonian with real eigenvalues, we define a new inner product on the Hilbert space with respect to which the non-Hermitian Hamiltonian is Quasi-Hermitian. The Quasi-hermiticity of the Hamiltonian introduces the bi-orthogonality between the left-hand eigenstates and the right-hand eigenstates, in which case the metric becomes a basis transformation. We use the non-Hermitian quadratic Hamiltonian to show that such a metric is not unique but can be uniquely defined by requiring to hermitize all elements of one of the irreducible sets defined on the set of all observables. We compare the constructed metric with specific known examples in the literature in which cases a unique choice is made. / AFRIKAANSE OPSOMMING: Ons ondersoek die implementering van n gepaste raamwerk virn nie-Hermitiese kwantumteorie. Ons beskoun nie-Hermitiese Hamilton-operator met reele eiewaardes en definieer in gepaste binneproduk ten opsigtewaarvan die operator kwasi-Hermitiese is. Die kwasi- Hermities aard van die Hamilton operator lei dan tot n stel bi-ortogonale toestande. Ons konstrueer n basistransformasie wat die linker en regter eietoestande van hierdie stel koppel. Hierdie transformasie word dan gebruik omn nuwe binneproduk op die Hilbert-ruimte te definieer. Die oorspronklike nie-HermitieseHamilton-operator is danHermitiesmet betrekking tot hierdie nuwe binneproduk. Ons gebruik die nie-Hermitiese kwadratieseHamilton-operator omte toon dat hierdie metriek nie uniek is nie, maar wel uniek bepaal kan word deur verder te vereis dat dit al die elemente van n onherleibare versameling operatoreHermitiseer. Ons vergelyk hierdie konstruksiemet die bekende voorbeelde in die literatuur en toon dat diemetriek in beide gevalle uniek bepaal kan word.

Thesis (MSc)--Stellenbosch University, 2003. / ENGLISH ABSTRACT: This thesis presents an overview of the flow equations recently introduced by Wegner. The little known mathematical framework is established in the initial chapter and used as a background for the entire presentation. The application of flow equations to the Foldy-Wouthuysen transformation and to the elimination of the electron-phonon coupling in a solid is reviewed. Recent flow equations approaches to the Lipkin model are examined thoroughly, paying special attention to their utility near the phase change boundary. We present more robust schemes by requiring that expectation values be flow dependent; either through a variational or self-consistent calculation. The similarity renormalization group equations recently developed by Glazek and Wilson are also reviewed. Their relationship to Wegner's flow equations is investigated through the aid of an instructive model. / AFRIKAANSE OPSOMMING: Hierdie tesis bied 'n oorsig van die vloeivergelykings soos dit onlangs deur Wegner voorgestel is. Die betreklik onbekende wiskundige raamwerk word in die eerste hoofstuk geskets en deurgans as agtergrond gebruik. 'n Oorsig word gegee van die aanwending van die vloeivergelyking vir die Foldy-Wouthuysen transformasie en die eliminering van die elektron-fonon wisselwerking in 'n vastestof. Onlangse benaderings tot die Lipkin model, deur middel van vloeivergelykings, word ook deeglik ondersoek. Besondere aandag word gegee aan hul aanwending naby fasegrense. 'n Meer stewige skema word voorgestel deur te vereis dat verwagtingswaardes vloei-afhanklik is; óf deur gevarieerde óf self-konsistente berekenings. 'n Inleiding tot die gelyksoortigheids renormerings groep vergelykings, soos onlangs ontwikkel deur Glazek en Wilson, word ook aangebied. Hulle verwantskap met die Wegner vloeivergelykings word bespreek aan die hand van 'n instruktiewe voorbeeld.

The Wiggins-Holmes extension of the generalized Melnikov method (GMM) is applied to weakly damped parametrically excited cross waves with surface tension in a long rectangular wave channel in order to determine if these cross waves are chaotic. The Lagrangian density function for surface waves with surface tension is simplified by transforming the volume integrals to surface integrals and by subtracting the zero variation integrals. The Lagrangian is written in terms of the three generalized coordinates (or, equivalently the three degrees of freedom) that are the time-dependent components of the velocity potential. A generalized dissipation function is assumed to be proportional to the Stokes material derivative of the free surface. The generalized momenta are calculated from the Lagrangian and the Hamiltonian is determined from a Legendre transformation of the Lagrangian. The first order ordinary differential equations derived from the Hamiltonian are usually suitable for the application of the GMM. However, the cross wave equations of motion must be transformed in order to obtain a suspended system for the application of the GMM. Only three canonical transformations that preserve the dynamics of the cross wave equations of motion are made because of an extension of the Herglotz algorithm to nonautonomous systems. This extension includes two distinct types of the generalized Herglotz algorithm (GHA). The system of nonlinear nonautonomous evolution equations determined from Hamilton's equations of motion of the second kind are averaged in order to obtain an autonomous system. The unperturbed system is analyzed to determine hyperbolic saddle points that are connected by heteroclinic orbits The perturbed Hamiltonian system that includes surface tension satisfies the KAM nondegeneracy requirements; and the Melnikov integral is calculated to demonstrate that the motion is chaotic. For the perturbed dissipative system with surface tension, the Melnikov integral is identically zero implying that a higher dimensional GMM is necessary in order to demonstrate by the GMM that the motion is chaotic. However, numerical calculations of the largest Liapunov characteristic exponent demonstrate that the perturbed dissipative system with surface tension is also chaotic. A chaos diagram is computed in order to search for possible regions of the damping parameter and the Floquet parametric forcing parameter where chaotic motions may exist. / Graduation date: 1999